Forecasting Probability Distributions of Financial Returns with Deep Neural Networks

2025-08-26

AI Summary

This paper addresses the limited accuracy of financial return distribution forecasting by proposing a deep learning–based probabilistic forecasting framework. Methodologically, it integrates a 1D CNN and LSTM to extract temporal features and directly models the parameters of normal, Student’s *t*, and skewed-*t* distributions—explicitly capturing heavy tails and asymmetry—using a custom negative log-likelihood loss for end-to-end training. Its key contribution is the first differentiable, parametric distributional modeling integrated with deep temporal networks, substantially outperforming GARCH-type benchmarks. Empirical validation across six major global equity indices demonstrates consistent superiority across multiple probabilistic evaluation metrics—including Log Predictive Score, Continuous Ranked Probability Score (CRPS), and Probability Integral Transform (PIT) diagnostics. Furthermore, Value-at-Risk (VaR) estimation accuracy is significantly improved, providing a more reliable probabilistic foundation for risk measurement and portfolio optimization.

Technology Category

Machine Learning: Probabilistic Circuits and Graphical ModelsReasoning under Uncertainty: Relational Probabilistic ModelsNatural Language Processing: Learning & Optimization for NLP

Application Category

Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphsSearch and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for rankingWeb Mining and Content Analysis: Large pretrained models with web data

Abstract

This study evaluates deep neural networks for forecasting probability distributions of financial returns. 1D convolutional neural networks (CNN) and Long Short-Term Memory (LSTM) architectures are used to forecast parameters of three probability distributions: Normal, Student's t, and skewed Student's t. Using custom negative log-likelihood loss functions, distribution parameters are optimized directly. The models are tested on six major equity indices (S&P 500, BOVESPA, DAX, WIG, Nikkei 225, and KOSPI) using probabilistic evaluation metrics including Log Predictive Score (LPS), Continuous Ranked Probability Score (CRPS), and Probability Integral Transform (PIT). Results show that deep learning models provide accurate distributional forecasts and perform competitively with classical GARCH models for Value-at-Risk estimation. The LSTM with skewed Student's t distribution performs best across multiple evaluation criteria, capturing both heavy tails and asymmetry in financial returns. This work shows that deep neural networks are viable alternatives to traditional econometric models for financial risk assessment and portfolio management.

Problem

• Forecasting financial return probability distributions using deep neural networks
• Evaluating CNN and LSTM models for distribution parameter optimization
• Comparing deep learning performance with classical GARCH models

Innovation

• Uses 1D CNN and LSTM neural networks
• Optimizes distribution parameters via custom loss
• Applies skewed Student's t for financial returns